Integration using Euler's formula

Integration using complex analysis is a method of integrating certain functions.

Suppose we wanted to integrate:

$\int e^{x}\cos xdx$

Instead of using Integration by parts, we may substitute the cosine function for its Euler form: $\cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}$

$\int e^{x}\cdot {\frac {e^{ix}+e^{-ix}}{2}}dx$

${1 \over 2}\int e^{x(1+i)}+e^{x(1-i)}dx$

This is far easier to integrate.

Alternatively, we may also take note of real and imaginary portions of complex numbers

Cosine is the real portion of a complex number written in cos x + i sin x form

$\int e^{x}\cos xdx=$

$\int e^{x}\mathrm {Re} \{\cos x+i\cdot \sin x\}dx$

$\int e^{x}\mathrm {Re} \{e^{ix}\}dx$

$\mathrm {Re} \{\int e^{x}e^{ix}dx\}$

$\mathrm {Re} \{\int e^{(i+1)x}dx\}$